DENSITY-FUNCTIONAL STUDY OF Zr-BASED ACTINIDE ALLOYS

23 Pages·2009·0.72 MB·English

LLNL-JRNL-410491
DENSITY-FUNCTIONAL STUDY
OF Zr-BASED ACTINIDE
ALLOYS: 2. U-Pu-Zr SYSTEM
A. Landa, P. Soderlind, P. Turchi, L. Vitos, A.
Ruban
February 11, 2009
Journal of Nuclear Materials
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Density-functional study of Zr-based actinide alloys:
2. U-Pu-Zr system

A. Landa1*, P. Söderlind1, P. E.A. Turchi1, L. Vitos2, and A. Ruban2
1Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
2Royal Institute of Technology, SE-10044, Stockholm, Sweden

Abstract
Density-functional theory, previously used to describe phase equilibria in the U-Zr alloys
[1], is applied to study ground state properties of the bcc U-Pu-Zr solid solutions.
Calculated heats of formation of the Pu-U and Pu-Zr alloys are in a good agreement with
CALPHAD assessments. We found that account for spin-orbit coupling is important for
successful description of Pu-containing alloys.

Keywords: nuclear reactor materials, phase transitions, first principles.

* Corresponding author. Tel.: + 1 925 424 3523; fax: [phone]. E-Mail address: [email protected]
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1. Introduction
In our previous paper, Ref. [1], we performed a detailed ab initio study of the
main thermodynamic properties of the U-Zr alloy system that is a good nuclear fuel
candidate for fast breeder reactors. Because the thermal conductivity of U-Zr alloys is
one of the highest among nuclear fuels, the radial temperature gradient generated in fast
breeder reactors appears to be sufficiently large to cause redistribution of the alloy
constituents [2, 3]. The constituent redistribution alters alloy composition during
irradiation, which affects local power production, variation in fission product generation,
and irradiation behavior such as fuel swelling and growth.
Although the U-Zr alloys can be used as nuclear fuel, a fast rectors operation on a
closed fuel cycle will, due to the nuclear reactions, contain significant amount of
plutonium [3]. The ternary U-Pu-Zr phase diagram is very complex, although the
constituent redistribution in the specific U-19Pu-10Zr (wt. %), or U Pu Zr (at. %),
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alloy was studied in numerous publications [2-6]. This alloy is known to form: (i) a bcc
solid solution ( -phase) in the temperature range of 650-750 °C; (ii) a mixture of - and
cubic - phases in the temperature range of 600-650 °C; and (iii) below 600 °C the -
phase transforms to the fcc -phase and a mixture of - and phase that exists in the
temperature range of 500-600 °C [4, 5]. A detailed study of the microstructure of the
irradiated U-19Pu-10Zr (wt. %) alloy [4-6] revealed three distinct concentric zones: the
pure -phase Zr-enriched porous central zone, the Zr-depleted intermediate zone ( +
mixture), and a slightly Zr-enriched zone on the periphery ( + mixture). Although Pu
concentration remains almost constant in all three zones with a slight decrease toward the
fuel surface, it is believed that Pu presence in the alloy at levels greater than 8 wt. %
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enhances U and Zr migration [3, 5]. Even though both irradiated U-Zr and U-Pu-Zr alloys
develop a porous Zr-enriched central and apparently dense Zr-depleted intermediate
zone, a key difference exists in their microstructure [3]. In the irradiated U-Pu-Zr fuel the
intermediate zone is almost completely depleted of Zr that migrates both radially inward
(to the central zone) and outward (to the periphery). In contrast, evidence of outward Zr
migration has not been found yet in the irradiated binary U-Zr fuel: the depleted
intermediate zone retains a fair amount of Zr, and the amount of Zr lost from the depleted
intermediate zone balances the amount gained in the enriched center [3]. The complex
nature of constituent redistribution in the U-Pu-Zr alloys during irradiation and its
influence on nuclear fuel performance makes study of this phenomenon extremely
important.
Recent semi-empirical model calculations [2], supported by experimental
observation, indicated that the excess enthalpy of solution of the bcc -U-Zr phase
controls the constituent redistribution process. In our previous paper [1], we performed
detailed calculations of the heat of formation of -U-Zr solid solutions. In this work we
present results of similar calculations for -Pu-U and -Pu-Zr alloys that can be used for
further analysis of the constituent redistribution in the central zone of U-Pu-Zr nuclear
fuels.
In our calculations we employ three complementary computational techniques: (i)
scalar-relativistic (SR) Green‟s function technique based on the Korringa-Kohn-Rostoker
(KKR) method within the atomic-sphere approximation (ASA), (ii) scalar-relativistic and
fully-relativistic (FR) exact muffin-tin orbital methods (EMTO), and (iii) the all-electron
full-potential linear muffin-tin orbital method (FPLMTO) that also accounts for all
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relativistic effects. Pertinent details of the computational methods are described in
Section 2. Results of the density-functional calculations of the ground-state properties of
the -Pu-U and -Pu-Zr solid solutions are presented in Section 3. Lastly, concluding
remarks are presented in Section 4.

2. Computational details
The calculations we have referred to as KKR-ASA are performed using the SR
Green‟s function technique based on the KKR method within the ASA [7-10]. For the
present study this approximation is improved by including higher multipoles of the
charge density [9] and the so-called muffin-tin correction [10] to the electrostatic energy.
The calculations are performed for a basis set including valence spdf orbitals. For the
electron exchange and correlation energy functional, the generalized gradient
approximation (GGA) is considered [11]. Integration over the Brillouin zone is
performed using the special k-point technique [12] with 506 points in the irreducible
wedge of the zone for the bcc structure. The moment of the density of states, needed for
the kinetic energy and valence charge density, are calculated by integrating the Green‟s
function over a complex energy contour (with a 2.5 Ry diameter) using a Gaussian
integration technique with 30 points on a semi-circle enclosing the occupied states. The
equilibrium density of the U-Zr, Pu-U, and Pu-Zr alloys is obtained from a Murnaghan
[13] fit to the total energy versus lattice constant curve.
In order to treat compositional disorder the KKR-ASA method is combined with
the coherent potential approximation (CPA) [14]. The ground-state properties of the
chemically random U-Zr, Pu-U, and Pu-Zr alloys are obtained from KKR-ASA-CPA
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calculations that include the Coulomb screening potential and energy [15-17]. The
screening constants are determined from supercell calculations using the locally self-
consistent Green‟s function (LSGF) method [18]. The and screening constants, see
Refs. [15,16] for details, are found to be 0.70 and 1.06, 0.93 and 1.17, and 0.57 and 0.81,
for U-Zr, Pu-U, and Pu-Zr alloys, respectively.
The SR-KKR-ASA-CPA formalism was very successful for describing the
ground-state properties of the U-Zr alloys [1], however the present study reveals that
relativistic effects are important for Pu-containing systems. Because of this we use the
Green‟s function technique, based on the EMTO formalism, which is not restricted to
specific geometries as imposed by the ASA, and also includes the spin-obit coupling
through the four-component Dirac equation [19].
The EMTO calculations are performed using both scalar-relativistic and full
relativistic Green‟s function techniques based on the improved screened KKR method,
where the one-electron potential is represented by optimized overlapping muffin-tin
(OOMT) potential spheres [20, 21]. Inside the potential spheres the potential is
spherically symmetric, and it is constant between the spheres. The radii of the potential
spheres, the spherical potentials inside the spheres, and the constant value in the
interstitial region are determined by minimizing (i) the deviation between the exact and
overlapping potentials, and (ii) the errors caused by the overlap between the spheres.
Within the EMTO formalism, the one-electron states are calculated exactly for the
OOMT potentials. As an output of the EMTO calculations, one can determine self-
consistent Green‟s function of the system and the complete, non-spherically symmetric
charged density. Finally, the total energy is calculated using the full charge-density
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technique [22]. Like in the case of KKR-ASA calculations, GGA is used for the electron
exchange and correlation approximation, and EMTO is combined with the CPA for the
calculation of the total energy of chemically random alloys [23]. Integrations over the
Brillouin zone and complex energy contour and the choice of the screening constants are
identical to those in the KKR-ASA method. Although spin-polarization was not
considered in the case of the U-Zr system (non-magnetic solution), the Pu-U and Pu-Zr
alloys have been modeled within the disordered local moment approximation that leads to
a paramagnetic solution, see Ref. [24] for details.
For the elemental metals, the most accurate and fully relativistic calculations are
performed using an all-electron approach where the relativistic effects, including spin-
orbit coupling, are accounted for. Although unable to model disorder in the CPA sense it
provides important information for the metals, and also serves to confirm the CPA
calculations mentioned above. For this purpose we use a version of the FPLMTO [25-
27]. The “full potential” in FPLMTO refers to the use of non-spherical contributions to
the electron charge density and potential. This is accomplished by expanding the charge
density and potential in cubic harmonics inside non-overlapping muffin-tin spheres and
in a Fourier series in the interstitial region. We use two energy tails associated with each
basis orbital, and for U‟s semi-core 6s, 6p states and valence states (7s, 7p, 6d, and 5f)
these pairs are different. With this „double basis‟ approach we use a total of six energy
tail parameters and a total of 12 basis functions per atom. Spherical harmonic expansions
are carried out up to l = 6 for the basis, potential, and charge density. As in the case of
max
the KKR-ASA and EMTO methods, GGA is used for the electron exchange-correlation
approximation. A special quasi-random structure (SQS) method, utilizing a 16-atom
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supercell, was used to treat the compositional disorder within the FPLMTO formalism
[28]. Spin polarization for the Pu-containing alloys was arranged in an antiferromagnetic
fashion [29] with neighboring atoms having anti-parallel spins. This is different from the
spin configuration used in the EMTO calculations.

3. Ground-state properties of -U-Pu-Zr solid solutions
Figure 1a shows results of SR-KKR-ASA-CPA calculations of the heat of
formation of the -U-Zr solid solutions at T = 0 K [1]. The heat of formation, that shows a
positive deviation from the energy associated with a mixture of the pure elements, agrees
well with the existence of a miscibility gap in the U-Zr phase diagram. Notice that the
calculated heat of formation of the -U-Zr solid solutions is in excellent agreement with
data derived from a CALPHAD assessment [30] of the experimental thermodynamics
and phase diagram information, which validates the ab initio approach. For comparison,
we also show the heats of formation for the U Zr , U Zr , and U Zr bcc alloys,
75 25 50 50 25 75
calculated within the FPLMTO-SQS technique that agrees pretty well with both SR-
KKR-ASA-CPA and CALPHAD assessment results.
Figure 1b show results of SR-KKR-ASA-CPA, SR-EMTO-CPA, and FR-EMTO-
CPA calculations of the heat of formation of the -Pu-U solid solutions at T = 0 K. Both
scalar-relativistic approaches (SR-KKR-ASA and SR-EMTO-CPA) reveal a positive heat
of formation that is in a stark contrast with CALPHAD and FPLMTO-SQS results also
shown in this Figure. Only when spin-orbit coupling is accounted for within the
framework of FR-EMTO-CPA formalism, the calculated heat of formation becomes
negative and in excellent agreement with CALPHAD results. In order to understand why
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the spin-orbit interaction plays an important role in describing the energetics of Pu-U
alloys, we present the calculated equilibrium atomic volumes of bcc Pu and U in the
Table. There is a reasonable agreement between the equilibrium atomic volume of bcc U
calculated by different approaches. This is not true in the case of bcc Pu, where both
KKRASA and SR-EMTO give a significantly larger equilibrium atomic volume than the
one calculated within the fully-relativistic approaches (FR-EMTO and FPLMTO). The f-
band occupation in Pu, calculated within the scalar-relativistic formalisms (SR-KKRASA
and SR-EMTO), which is also presented in this Table (in the parenthesis), is about 5.4
that is higher than the 5.1–5.2 calculated when spin-orbit coupling is taken into
consideration within the FR-EMTO and FPLMTO methods. The later values are in
excellent agreement with the photoelectron spectroscopy estimation [31] as well as recent
DMFT results [32]. The present study allows us to conclude that spin-orbit coupling
becomes crucially important for calculating the excess thermodynamic characteristics of
Pu-containing alloys.
Figure 1c show results of SR-KKR-ASA-CPA, SR-EMTO-CPA, FR-EMTO-
CPA, and FPLMTO-SQS calculations for the heat of formation of the -Pu-Zr solid
solutions at T = 0 K. All CPA-related models reveal a change in the sign of the heat of
formation as a function of alloy composition with a tendency towards phase formation for
Zr-rich alloys. However this energetic evolution is in contrast with the CALPHAD
results. One plausible explanation is that the CALPHAD results were obtained from data
for the high-temperature part of the Pu-Zr phase diagram. Therefore it is likely that the -
phase is primarily stabilized by entropy. It is also worth noting that in all cases the
extremum heat of formation is small (note the energy scale compared to the one for U-Zr
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